RelatingLength,SurfaceArea,andVolume

　　The question asks for the ratio between the radii of the sphere and the cylinder. This ratio is given by r s/rc. Now you can solve the equation 4πrs2 = πrc2　h for the ratio rs/rc.

　　Changing Measurements　　The second way the Math IC will test your understanding of the relationships among length, surface area, and volume is by changing one of these measurements by a given factor, and then asking how this change will influence the other measurements.　　When the lengths of a solid in the question are increased by a single constant factor, a simple rule can help you find the answer:If a solid’s length is multiplied by a given factor, then the solid’s surface area is multiplied by the square of that factor, and its volume is multiplied by the cube of that factor.　　Remember that this rule holds true only if all of a solid’s dimensions increase in length by a given factor. So for a cube or a sphere, the rule holds true when just a side or the radius changes, but for a rectangular solid, cylinder, or other solid, all of the length dimensions must change by the same factor. If the dimensions of the object do not increase by a constant factor—for instance, if the height of a cylinder doubles but the radius of the base triples—you will have to go back to the equation for the dimension you are trying to determine and calculate by hand.　　Example 1

　　If you double the length of the side of a square, by how much do you increase the area of that square?

　　If you understand the formula for the area of a square, this question is simple. The formula for the area of a square is A = s2, where s is the length of a side. Replace s with 2s, and you see that the area of a square quadruples when the length of its sides double: (2s)2 = 4s2.　　Example 2

　　If a sphere’s radius is halved, by what factor does its volume decrease?

　　The radius of the sphere is multiplied by a factor of 1 2 (or divided by a factor of 2), and so its volume multiplies by the cube of that factor: (1 2)3 = 1 8. Therefore, the volume of the sphere is multiplied by a factor of 1 8 (divided by 8), which is the same thing as decreasing by a factor of 8.　　Example 3

　　A rectangular solid has dimensions x　y　z (these are its length, width, and height), and a volume of 64. What is the volume of a rectangular solid of dimensions x /2　y /2　z?

　　If this rectangular solid had dimensions that were all one-half as large as the dimensions of the solid whose volume is 64, then its volume would be (1 2)3　64 = 1 8　64 = 8. But dimension z is not multiplied by 1 2 like x and y. To answer a question like this one, you should use the volume formula for rectangular solids: Volume = l　w　h. It is given in the question that xyz = 64. So, x 2　y 2　z = 1 4　xyz = 1 4　64 = 16.

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